GENIUS AND POVERTY of assuming that people are depraved because they drink to excess, Galton inverted this hypothesis and assumed temporarily that people drink to excess because they have inherited no moral guts from their ancestors, in short, because they are depraved. Brushing aside all the vaporous moralizing of the reformers, Galton took a firm grip on a scientific, unemotional, workable hypothesis to which he could apply the impartial machinery of mathematics. His work has not yet registered socially. For the moment we need note only that Galton, like Abel, inverted his problem - turned it upside-down and inside-out, back-end-to and foremost-end-backward. Like Hiawatha and his fabulous mittens, Galton put the skinside inside and the inside outside. All this is far from being obvious or a triviality. It is one of the most powerful methods of mathematical discovery (or invention) ever devised, and Abel was the first human being to use it consciously as an engine of research. 'You must always invert', as Jacobi said when asked the secret of his mathema- tical discoveries. He was recalling what Abel and he had done* If the solution of a problem becomes hopelessly involved, try turning the problem backwards, put the quaesita for the data and vice versa. Thus if we find Cardan's character incompre- hensible when we think of him as a son of his father, shift the emphasis, invert it, and see what we get when we analyse Cardan's father as the begetter and endower of his son. Instead of studying 'inheritance' concentrate on 'endowing'. To return to those who remember some trigonometry. Suppose mathematicians had been so blind as not to see that sin #, cos # and the other direct trigonometric functions are simpler to use, in the addition formulae and elsewhere, than the inverse functions sin-1 a, cos-1 02. Recall the formula sin (x + y) in terms of sines and cosines of x and y, and contrast it with the formula for sin"1 (x + y) in terms of # and y. Is not the former incomparably simpler, more elegant, more 'natural' than the latter? Now, in the integral calculus, the inverse trigonometric functions present themselves naturally as definite integrals'of simple algebraic irrationalities (second degree); such integrals appear when we seek to find the length of an arc of a circle by means of the integral calculus. Suppose the inverse trigono- 355